Many applications feature autocorrelated count data \(X_t\)  at discrete times
\(t\) . A number of authors have constructed and studied stationary stochastic
processes \(X_t\)  whose one-dimensional marginal distributions come from an
arbitrary infinitely-divisible distribution family \(\lbrace \mu ^{\theta }\rbrace \) , such
as the Poisson \(\textsf {Po}(\theta )\)  or negative binomial \(\textsf {NB}(\theta ,p)\) , and that
are “\(\textsf {AR}(1)\) -like” in the sense that their autocorrelation function is
\({\textsf {Corr}}[X_s,X_t] =\rho ^{|s-t|}\)  for some \(\rho \in (0,1)\)  [1]}, [2]}, [3]}, [4]}, [5]}. The most common
approach is to build a time-reversible Markov process using thinning,
in which the process at any two consecutive times may be written in the form
\( X_{t-1}=\xi _t+\eta _t \qquad X_t=\xi _t+\zeta _t \) 
